In this form, we can describe the general situation. A linear transformation T : V W is called an isomorphism if it is both onto and one-to-one. The vector spaces V and W are said to be → …

Fact: every isomorphism is both a section and a retraction.

11In other words, a homomorphism will commute with multiplication in that they can be applied in either order. This results in a commutative diagram.

ISOMORPHISM 1. One to One Correspondence If f is a function from f1; 2; 3g to f4; 5; 6g; we often summarize its domain and target sets by the notation : f1; 2; 3g ! f4; 5; 6g:

rite G = eG. An isomorphism between two groups is a dictionary that lets us translate elements and operations from one group to the other without losing essential.

We're studying vector spaces, so we need a pre-cise de nition of isomorphism for them. e nition 1 (Isomorphism of vector spaces). Two vector spaces V and W over the same eld F are isomorphic if …